{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:U7X33EJNIJSVCEFRYXHKBHPMPJ","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"c04fb7adc57349898edf0bf31bfb526add179eb880176a455b7ef2fa57ec27e6","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2012-07-26T18:50:00Z","title_canon_sha256":"c0d51014f56cee95ab3533dcd93b5ea2534acb9a705c073060fd9556e30791bd"},"schema_version":"1.0","source":{"id":"1207.6365","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1207.6365","created_at":"2026-05-18T03:28:56Z"},{"alias_kind":"arxiv_version","alias_value":"1207.6365v4","created_at":"2026-05-18T03:28:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1207.6365","created_at":"2026-05-18T03:28:56Z"},{"alias_kind":"pith_short_12","alias_value":"U7X33EJNIJSV","created_at":"2026-05-18T12:27:23Z"},{"alias_kind":"pith_short_16","alias_value":"U7X33EJNIJSVCEFR","created_at":"2026-05-18T12:27:23Z"},{"alias_kind":"pith_short_8","alias_value":"U7X33EJN","created_at":"2026-05-18T12:27:23Z"}],"graph_snapshots":[{"event_id":"sha256:e6ff6698de1be0fc31e10107d699ef387b00517bd24a91bb77555a22ca668cfa","target":"graph","created_at":"2026-05-18T03:28:56Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We design a new distribution over $\\poly(r \\eps^{-1}) \\times n$ matrices $S$ so that for any fixed $n \\times d$ matrix $A$ of rank $r$, with probability at least 9/10, $\\norm{SAx}_2 = (1 \\pm \\eps)\\norm{Ax}_2$ simultaneously for all $x \\in \\mathbb{R}^d$. Such a matrix $S$ is called a \\emph{subspace embedding}. Furthermore, $SA$ can be computed in $\\nnz(A) + \\poly(d \\eps^{-1})$ time, where $\\nnz(A)$ is the number of non-zero entries of $A$. This improves over all previous subspace embeddings, which required at least $\\Omega(nd \\log d)$ time to achieve this property. We call our matrices $S$ \\emp","authors_text":"David P. Woodruff, Kenneth L. Clarkson","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2012-07-26T18:50:00Z","title":"Low Rank Approximation and Regression in Input Sparsity Time"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.6365","kind":"arxiv","version":4},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:35ae4df8f155f23bea223fe86e9b26f6bf7007edf26894b70d2ad57dbc821671","target":"record","created_at":"2026-05-18T03:28:56Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"c04fb7adc57349898edf0bf31bfb526add179eb880176a455b7ef2fa57ec27e6","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2012-07-26T18:50:00Z","title_canon_sha256":"c0d51014f56cee95ab3533dcd93b5ea2534acb9a705c073060fd9556e30791bd"},"schema_version":"1.0","source":{"id":"1207.6365","kind":"arxiv","version":4}},"canonical_sha256":"a7efbd912d42655110b1c5cea09dec7a7986c120c0a1f8bc70f236cb11e11868","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a7efbd912d42655110b1c5cea09dec7a7986c120c0a1f8bc70f236cb11e11868","first_computed_at":"2026-05-18T03:28:56.285489Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:28:56.285489Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"zMys5Dl7WSIcPQVaMjrMgzrdy6gwNsTXDL9auF3aN5Ya+RqMiiaSDTyYKLGuFpWsDZNHTpayuPPe1xpRmzChBw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:28:56.286235Z","signed_message":"canonical_sha256_bytes"},"source_id":"1207.6365","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:35ae4df8f155f23bea223fe86e9b26f6bf7007edf26894b70d2ad57dbc821671","sha256:e6ff6698de1be0fc31e10107d699ef387b00517bd24a91bb77555a22ca668cfa"],"state_sha256":"8197bd78e9b43cb6b3542e61b97a1d3c0926418bbd7ff40eeaa38328f306b79e"}